Chapter 1 Late 1800 s Several failures of classical (Newtonian) physics discovered

Transcription

1 Chaptr 1 Lat 1800 s Svral failurs of classical (Nwtonian) physics discovrd Dvlopmnt of QM rsolvd discrpancis btwn xpt. and classical thory QM Essntial for undrstanding many phnomna in Chmistry, Biology, Physics photosynthsis + vision (lctron xcitation) vibrations/rotations (xcitation of nuclar motion) magntic rsonanc imaging radioactivity opration of transistors lasrs (CD + DVD playrs) van dr Waals intractions (from M. Johnson, Yal Univrsity)

2 Problms whr classical physics is inadquat 1. Blackbody Radiation hatd objcts light Classical thory (Rayligh- Jans (1905) Planck: (1900) spctral dnsity = 8 d E 3 oscd c 8 ktd 3 c Emits nrgy at all T d 3 8h 1 3 c h d 1 = frquncy = spctral dnsity (nrgy volum -1 frquncy -1 ) E OSC = avrag nrgy/osc k = Boltzmann const. c = spd of light originally dtrmind by fitting xprimnt

3 Planck's constant: 34 h 6.66x10 Js Planck latr showd that his xprssion is consistnt with th nrgis of th oscillators making up th blackbody objct taking on discrt valus E nh, n 0,1,,... E h h 1 T 0 0 T kt Classical rsult Bottom rsult obtaind using Taylor sris of x for small x: x 1 x...

4 Drivation of Planck s rsult Possibl nrgis: 0, h, h, 3h, tc. Probability of having nrgy nh givn by E / EkT n E p n = Boltzmann factor n0 nh n0 nh nh p n n nh nh normalization (sum ovr all lvls) h h 1 1 dnominator = xx... 1x 1 h E numrator = h h/ kt 1 1 h h h 3 h/ kt h h h h/ kt h 1 h h h h h avg. nrgy of mod of frq. h

5 . Photolctric ffct light - mtal xpctd bhavior light is a wav, so ach - absorbs small fraction of th nrgy - mittd at all, if intnsity (I) grat nough KE of jctd - > with > intnsity obsrvd # - mittd intnsity - mittd if > 0 (critical frq.) KE > with >, and indpndnt of intnsity

6 Explaind by Einstin in 1905 light has nrgy h (E = h) and acts particl-lik, nabling its nrgy to b focusd on on - nrgy of jctd lctron = E l = h -, = work function of mtal (~IP) critical frquncy (implis critical nrgy) for production of photolctrons

7 3. Hat capacity of solids (classical = 3R; actual 0 as T 0) (R = N a k) Classical rsult Cv 3Nk E 3NkT Expt. Einstin s solution: Assumd vibrations of lattic ar quantizd, i.., nrgis ar nh Figur from Wikipdia

8 4. Wav-particl duality photolctric ffct light can bhav as a particl diffraction of light light can bhav as a wav d Brogli (194): particls hav a wavlngth: Dmonstratd by diffraction of -, H, H from crystallin surfacs - with KE = 17 V has = 3 Å, a typical lattic spacing in a crystal intrfrnc (diffraction) larg objcts basballs, cars, tc., hav d Brogli wavlngths too small to b dtctd h p Basball wighs ~0.14 kg; 100mph = 44.7m/s λ = 1.05x10-39 m

9 Diffraction xprimnts light incidnt on a singl slit n sin, a minima: n = ±1, ±, ±3, a: wll sparatd paks >> a: can t s diffraction doubl-slit xpt. with - th - gos through both slits!! In 1977 th xpt. was don with th H atoms Each atom gos through both slits!! Commnt on singl flashs on scrn (forcs - back to particl natur)

10 5. Spctra of atoms + molculs discrt lins Figur from Wikipdia

11 IP spctrum H atom R H 1 1 n n 1 Rydbrg sris n 1, n intgrs, n = n 1 +1, n 1 +, n 1 +3, R H = 109, cm -1 (1 V = 8066 cm -1 ) 13.6 V Explaind by Bohr modl (1911) 4 r o Coulomb forc mv r cntrifugal forc + -

12 h r n n p nh mvr n Bohr basd his rsult on th corrspondnc Principl, rathr than th d Brogli rlation, which wasn t proposd until svral yars latr.. S plato.stanford.du/ntris/bohr~corrspondnc/) Valid solutions: th wav prfctly fits in th orbit r 4 0 m n E E E tot kintic pot m, n 1,, r 0h n v n mr 4 r r 0 mn 3 m r 4 0n m givs th dfinition of Rydbrg s constant

13 Summary: nrgy + oscillators ar quantizd wav-particl duality d Brogli rlationship ths idas pavd th way for QM NOTE: Frquncis of a guitar string ar quantizd (and guitars ar clarly Classical) Quantization coms from boundary conditions Fourir transforms: (frquncy tim) (position momntum) ar conjugat variabls W will com back to ths considrations.